Chapter 1.3, Problem 46ES

These exercises develop some versions of the substitution principle, a useful tool for the evaluation of limits.

(a) Suppose $g\left(x\right)\to -\infty$ as $x\to +\infty$ . Given any function $f\left(x\right)$ , explain why we can evaluate ${\lim }_{x\to +\infty }f\left[g\left(x\right)\right]$ by substituting $t=g\left(x\right)$ and writing

$\underset{x\to +\infty }{\lim }f\left[g\left(x\right)\right]=\underset{t\to -\infty }{\lim }f\left(t\right)$

(Here, “equality� is interpreted to mean that either both limits exist and are equal or that both limits fail to exist.)

(b) Why does the result in part (a) remain valid if ${\lim }_{x\to +\infty }$ is replaced everywhere by one of ${\lim }_{x\to -\infty },{\lim }_{x\to c},{\lim }_{x\to c^{-}}$ , or ${\lim }_{x\to c^{+}}$ ?